Singular value decomposition (svd) in MATLAB - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of SVD on a simple matrix

What is the output of the following MATLAB code snippet?

MATLAB

A = [3 1; 1 3];
[U,S,V] = svd(A);
disp(diag(S)');

A [2 4]

B [4 2]

C [5 1]

D [3 3]

💡 Hint

Recall that singular values are the square roots of eigenvalues of A'*A.

The matrix A'*A has eigenvalues 16 and 4, so singular values are sqrt(16)=4 and sqrt(4)=2.

🧠 Conceptual

intermediate

Properties of matrices in SVD

Which statement about the matrices U, S, and V from the SVD of a matrix A is correct?

A U and V are orthogonal matrices, and S is a diagonal matrix with non-negative entries.

B U and V are diagonal matrices, and S is an orthogonal matrix.

C U is upper triangular, V is lower triangular, and S is symmetric.

D U and V are symmetric matrices, and S is a diagonal matrix with negative entries.

💡 Hint

Think about the definition of orthogonal matrices and singular values.

In SVD, U and V are orthogonal matrices (U'*U=I, V'*V=I), and S is diagonal with non-negative singular values.

🔧 Debug

advanced

Identify the error in SVD reconstruction

What is the error in the following MATLAB code that attempts to reconstruct matrix A from its SVD?

MATLAB

A = [1 2; 3 4];
[U,S,V] = svd(A);
A_reconstructed = U * S * V;
disp(A_reconstructed);

A There is no error; the code reconstructs A correctly.

B The svd function returns V transposed, so V should be transposed again.

C The multiplication order is incorrect; it should be U * S * V'.

D The matrix S is not diagonal, so reconstruction fails.

💡 Hint

Recall the formula for reconstructing A from U, S, and V.

The correct reconstruction is A = U * S * V', not U * S * V.

📝 Syntax

advanced

Syntax error in SVD usage

Which option contains a syntax error when performing SVD in MATLAB?

A [U,S,V] = svd A;

B [U,S,V] = svd(A);

C [U,S,V] = svd(A, 'econ');

D [U,S,V] = svd(A,0);

💡 Hint

Check the correct syntax for function calls in MATLAB.

In MATLAB, function arguments must be enclosed in parentheses; 'svd A;' is invalid syntax.

🚀 Application

expert

Number of singular values greater than a threshold

Given matrix A, how many singular values are greater than 1.5 in the following code?

MATLAB

A = [4 0 0; 0 3 0; 0 0 0.5];
[~,S,~] = svd(A);
sv = diag(S);
count = sum(sv > 1.5);

A 0

B 3

C 1

D 2

💡 Hint

Look at the diagonal entries of S and count those greater than 1.5.

The singular values are 4, 3, and 0.5; only 4 and 3 are greater than 1.5, so count is 2.

Practice

1. What does the MATLAB function svd(A) return when applied to a matrix A?

easy

A. The determinant of matrix A.

B. The inverse of matrix A.

C. Three matrices: U, S, and V such that A = U * S * V'.

D. The transpose of matrix A.

Solution

1. Step 1: Understand the purpose of svd

   The svd function decomposes a matrix into three parts: U, S, and V.

2. Step 2: Recall the decomposition formula

   It satisfies the equation A = U * S * V', where U and V are orthogonal matrices and S is diagonal.

3. Final Answer:

   Three matrices: U, S, and V such that A = U * S * V'. -> Option C

4. Quick Check:

   svd(A) = U, S, V [OK]

Hint: Remember svd splits matrix into U, S, V parts [OK]

Common Mistakes:

• Confusing svd with inverse or transpose
• Expecting a single matrix output
• Thinking svd returns determinant

2. Which of the following is the correct MATLAB syntax to compute the singular value decomposition of matrix A and store the results in U, S, and V?

easy

A. [U, S, V] = svd(A);

B. svd(U, S, V) = A;

C. U, S, V = svd(A);

D. svd(A, U, S, V);

Solution

1. Step 1: Recall MATLAB function output syntax

   In MATLAB, multiple outputs are assigned using square brackets on the left side.

2. Step 2: Match correct svd syntax

   The correct syntax is [U, S, V] = svd(A); to get all three matrices.

3. Final Answer:

   [U, S, V] = svd(A); -> Option A

4. Quick Check:

   Use square brackets for multiple outputs [OK]

Hint: Use square brackets for multiple outputs in MATLAB [OK]

Common Mistakes:

• Using parentheses instead of brackets for outputs
• Trying to assign outputs on the right side
• Incorrect argument order in svd

3. Given the MATLAB code:

A = [3 1; 1 3];
[U, S, V] = svd(A);
disp(diag(S));

What will be the output of disp(diag(S))?

medium

A. [0; 0]

B. [3; 1]

C. [1; 3]

D. [4; 2]

Solution

1. Step 1: Calculate singular values of A

   Matrix A is symmetric. Its singular values are the absolute values of eigenvalues. Eigenvalues of A are 4 and 2.

2. Step 2: Understand svd output

   The diagonal of S contains singular values in descending order: 4 and 2.

3. Final Answer:

   [4; 2] -> Option D

4. Quick Check:

   Singular values = eigenvalues for symmetric matrix [OK]

Hint: For symmetric matrices, singular values = absolute eigenvalues [OK]

Common Mistakes:

• Confusing singular values with matrix entries
• Mixing order of singular values
• Expecting singular values to be zero

4. Consider the MATLAB code snippet:

A = [1 2; 3 4];
[U, S, V] = svd(A);
U = svd(A);

What is the error in this code?

medium

A. Matrix A is not square, so svd cannot be applied.

B. The second call to svd assigns a matrix to U instead of three outputs.

C. The variable V is not assigned in the second svd call.

D. The svd function requires four output variables.

Solution

1. Step 1: Analyze the first svd call

   The first call correctly assigns three outputs: U, S, V.

2. Step 2: Analyze the second svd call

   The second call assigns U = svd(A); which returns only singular values as a vector, not a matrix U.

3. Final Answer:

   The second call to svd assigns a matrix to U instead of three outputs. -> Option B

4. Quick Check:

   svd(A) alone returns singular values vector [OK]

Hint: svd(A) alone returns singular values vector, not matrices [OK]

Common Mistakes:

• Expecting svd(A) to return U matrix alone
• Thinking svd requires square matrix only
• Ignoring number of outputs in function call

5. You want to approximate a matrix A by keeping only the largest singular value and corresponding vectors using MATLAB. Which code snippet correctly performs this low-rank approximation?

hard

A. [U, S, V] = svd(A);A_approx = U(:,1) * S(1,1) * V(:,1)';

B. [U, S, V] = svd(A);A_approx = U(:,1) * V(:,1)';

C. [U, S, V] = svd(A);A_approx = S(1,1) * V(:,1)';

D. [U, S, V] = svd(A);A_approx = U(:,1) * S(:,1) * V(:,1)';

Solution

1. Step 1: Understand low-rank approximation using svd

   Keeping only the largest singular value means using the first column of U, the first singular value, and the first column of V.

2. Step 2: Construct the approximation

   The approximation is A_approx = U(:,1) * S(1,1) * V(:,1)', multiplying the first singular value and vectors.

3. Final Answer:

   [U, S, V] = svd(A); A_approx = U(:,1) * S(1,1) * V(:,1)'; -> Option A

4. Quick Check:

   Low-rank approx = U(:,1)*S(1,1)*V(:,1)' [OK]

Hint: Multiply first U column, singular value, and V column for approximation [OK]

Common Mistakes:

• Omitting singular value in multiplication
• Using entire S matrix instead of first singular value
• Incorrect matrix dimensions in multiplication